An Improved FPTAS for 0-1 Knapsack

04/21/2019

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by   Ce Jin, et al.

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The 0-1 knapsack problem is an important NP-hard problem that admits fully polynomial-time approximation schemes (FPTASs). Previously the fastest FPTAS by Chan (2018) with approximation factor 1+ε runs in Õ(n + (1/ε)^12/5) time, where Õ hides polylogarithmic factors. In this paper we present an improved algorithm in Õ(n+(1/ε)^9/4) time, with only a (1/ε)^1/4 gap from the quadratic conditional lower bound based on (,+)-convolution. Our improvement comes from a multi-level extension of Chan's number-theoretic construction, and a greedy lemma that reduces unnecessary computation spent on cheap items.